The speakers are clearly the weakest link and always have the biggest impact on the final sound. So I would spend the full budget on those, unless you also want to go into streaming audio.
As for speakers, how about the Harbeth SuperHL5+? I am not sure how much these would cost in the US, but I guess they would deplete much of the budget (they are worth it). Don’t waste any money on snake oil like cables. However, as others have said, do invest a bit of money into a calibrated measurement microphone like the UMIK-1 and familiarize yourself with the free REW software to analyze room response. Speakers are the weakest link in any system, and their in-room response is worse again.
Your amplifier should be fine, although at some time in the future you might want to opt for more power, depending on your listening habits.
If you want to try streaming, a Chromecast Audio is a cheap ($35) and remarkably good entry to discover if you like the idea. The inbuilt DAC is fine for 16/44 (i.e. cd) quality, but if you want more it also has an optical digital output (but you would need a DAC for that).
If at some stage you want to integrate your video system with your stereo system (a smart move), an Oppo 205 UHD Bluray player is all you need to play discs, and is an excellent external DAC for streaming.